MEMS structure stress distribution analysis and optimization method in micro-nano processing technology

Abstract

The invention discloses an MEMS structure stress distribution analysis and optimization method in a micro-nano processing technology, and relates to the technical field of optimization design. The method comprises the following steps: step 1, determining geometric parameters and material parameters of each layer of structure of the MEMS multi-layer composite structure, and constructing a stress field of the MEMS multi-layer composite structure; based on a piezoelectric elastic coupling effect, according to the stress field, establishing a piezoelectric equation; according to the strain tensor, based on a thermal coupling effect, solving a thermal coupling equation to obtain temperature distribution; 2, correcting the stress field according to the temperature distribution to obtain a corrected stress field; performing fatigue evaluation by using a preset fatigue accumulation evaluation model to obtain a fatigue damage accumulation coefficient; and step 3, setting constraint conditions according to the step 1 and the step 2, constructing an objective function according to the fatigue damage accumulation coefficient, and solving the objective function through iteration. The stability, the reliability and the service life of the MEMS device in a complex working environment can be effectively improved.

Description

Technical Field

[0001] The present invention relates to the technical field of optimization design, and particularly to a method for analyzing and optimizing the stress distribution of MEMS structures in micro-nano processing technology. Background Art

[0002] With the development of micro-nano processing technology, micro-electro-mechanical systems (MEMS) have been widely used in the fields of sensors, actuators, biomedicine, and aerospace. MEMS devices, with their characteristics of small size, high precision, and low power consumption, have become a key component of modern industrial and technological innovation. However, as MEMS devices develop towards multi-functionalization and high-performance, the design and manufacturing of their internal multi-layer composite structures have become increasingly complex. In order to improve the reliability and service life of MEMS devices, the research on stress distribution, fatigue accumulation, and thermo-mechanical coupling effects in multi-layer composite structures has become the focus of the current technical field. However, there are still many deficiencies in the research and solutions to these problems in the prior art.

[0003] Currently, the disclosed MEMS structure design technologies usually focus on static stress analysis and neglect the dynamic multi-field coupling effects in complex working environments. Traditional MEMS stress analysis methods are typically based on finite element analysis, which models the stress fields of single materials or single-layer structures to obtain preliminary results of stress distributions. For example, the disclosed finite element simulation technologies can analyze the maximum stress and strain distributions in structures under static loads, providing references for design optimization. However, simplified model assumptions are often used in traditional methods, such as neglecting the size effects, surface effects, and interface effects of materials. These simplifications lead to a significant reduction in the accuracy of analysis results at the micro-nano scale, making it difficult to capture real stress concentration phenomena and changes in material properties. Additionally, in the existing technologies, there is still relatively little research on the fatigue accumulation behavior in multi-layer composite structures. Some of the disclosed methods are mostly based on traditional fatigue theories, such as Miner's rule, which simply regards the fatigue damage accumulation of materials as a linear superposition of stress amplitudes and cycle numbers. However, Miner's rule fails to consider the non-linear effects of microstructures, interface shear stresses, and temperature field changes on fatigue accumulation behavior. This limitation makes it difficult for traditional methods to accurately predict the fatigue life of MEMS devices under cyclic loads. Especially in multi-layer composite structures, the interface effects between different material layers play a crucial role in the fatigue accumulation process, and there is generally a lack of quantitative analysis of interface shear stresses and interface failure mechanisms in the existing technologies. Moreover, there are still significant deficiencies in the existing technologies when dealing with the thermo-mechanical coupling effects in MEMS multi-layer composite structures. MEMS devices usually operate in complex thermal environments, such as high temperatures, high frequencies, or conditions with drastic temperature changes. These thermal environments can cause uneven expansion or contraction of materials, leading to stress redistribution and interface failure. Most current thermodynamics analysis methods adopt a single thermal expansion model, which can only describe the linear thermo-stress behavior of materials and neglect the dynamic effects of temperature gradients and heat dissipation on the stress field. This simplified model is particularly insufficient when dealing with non-uniform temperature fields at the micro-nano scale and cannot effectively predict the phenomena of local stress concentration and aggravated fatigue damage. Summary of the Invention

[0004] The objective of the present invention is to provide a method for analyzing and optimizing the stress distribution of MEMS structures in micro-nano processing technologies, which can effectively improve the stability, reliability, and service life of MEMS devices in complex working environments, providing strong technical support for the design of sensors, actuators, and other high-performance MEMS devices.

[0005] To solve the above technical problems, the present invention provides a method for analyzing and optimizing the stress distribution of MEMS structures in micro-nano processing technologies, and the method includes:

[0006] Step 1: Determine the geometric parameters and material parameters of each layer structure of the MEMS multi-layer composite structure, and construct the stress field of the MEMS multi-layer composite structure; based on the piezoelectric elastic coupling effect, according to the stress field, establish the piezoelectric equation, and solve to obtain the strain tensor; according to the strain tensor, based on the thermal-mechanical coupling effect, establish the thermal-mechanical coupling equation, and solve the thermal-mechanical coupling equation to obtain the temperature distribution;

[0007] Step 2: According to the temperature distribution, correct the stress field to obtain the corrected stress field; according to the corrected stress field, use the preset fatigue cumulative evaluation model to conduct fatigue evaluation to obtain the fatigue damage accumulation coefficient;

[0008] Step 3: Set the constraint conditions according to Step 1 and Step 2, construct the objective function according to the fatigue damage accumulation coefficient, and obtain the optimal solution of the objective function through iterative solution of the objective function; the optimal solution of the objective function includes the optimal geometric parameters and optimal material parameters of the MEMS multi-layer composite structure; use the optimal geometric parameters and optimal material parameters to optimize the design of the MEMS structure in the micro-nano processing technology.

[0009] Further, the geometric parameters of each layer structure include: layer thickness, material characteristic length, and stress attenuation length; the material parameters of each layer structure include: Young's modulus, Poisson's ratio, surface energy, and interfacial shear stress.

[0010] Further, in Step 1, the stress field of the MEMS multi-layer composite structure is expressed by the following formula:

[0011]

[0012] where, Q n is the Young's modulus of the nth layer structure; v n is the Poisson's ratio of the nth layer structure; γ n is the surface energy of the nth layer structure; h n is the layer thickness of the nth layer structure; l0 is the material characteristic length, which is a preset value; l c is the stress attenuation length, which is a preset value; τ int (x, y) is the interfacial shear stress at the position (x, y); x is the X-axis coordinate of the two-dimensional plane; y is the Y-axis coordinate of the two-dimensional plane; z is the Z-axis coordinate of the three-dimensional space; z int is the interface position; δ(·) is the Dirac function; σ total is the stress field; N is the number of layers of the MEMS multi-layer composite structure.

[0013] Further, the piezoelectric equation in Step 1 is expressed by the following formula:

[0014]

[0015] Among them, i, j, k, l, and m are all subscript indices representing directions, and their values are all 1 or 2; when the values of i and j are both 1, it represents the normal strain in the X-axis direction; when the value of i is 1 and the value of j is 2, it represents the shear strain in the two-dimensional plane formed by the X-axis and the Y-axis; when the values of k and l are both 1, it represents the normal stress in the X-axis direction; when the value of k is 1 and the value of l is 2, it represents the shear stress in the two-dimensional plane formed by the X-axis and the Y-axis; when the value of m is 1, the electric displacement is in the X-axis direction; when the value of m is 2, the electric displacement is in the Y-axis direction; D m is the electric displacement vector; S ijkl is the compliance coefficient; ε ij is the strain tensor, which is a symmetric quantity, ε ij = ε ji ; d mij are all piezoelectric strain coefficients; represents the divergence of D m ; represents the second-order gradient of σ total ; α cp is the micro-nano scale correction coefficient, which is a preset value; ∈ mk represents the dielectric coefficient; E m is the electric field strength vector.

[0016] Furthermore, the thermo-mechanical coupling equation in step 1 is calculated using the following formula:

[0017]

[0018] Among them, ρ is the density of the MEMS multi-layer composite structure; c p is the specific heat capacity of the MEMS multi-layer composite structure; T is the temperature distribution; t is the time; · represents the divergence calculation; represents the gradient of the temperature distribution; κ is the thermal conductivity of the MEMS multi-layer composite structure; α T is the preset thermo-mechanical coupling coefficient; η is the preset energy dissipation coefficient; M is the mass of the MEMS multi-layer composite structure.

[0019] Furthermore, in step 2, according to the following formula, the stress field is corrected based on the temperature distribution to obtain the corrected stress field σ modified :

[0020]

[0021] Among them, T0 is the initial temperature; λ th is the preset thermo-mechanical sensitivity coefficient.

[0022] Further, in step 2, according to the following formula, using a preset fatigue cumulative assessment model, fatigue assessment is performed based on the corrected stress field to obtain the fatigue damage accumulation coefficient D:

[0023]

[0024] where b is the preset material fatigue index. The smaller the value of b, the more easily the MEMS multi-layer composite structure is affected by stress and fatigued; σ′ f is the fatigue strength constant of the preset MEMS multi-layer composite structure, and its value range is from 20 to 30; W s is the elastic strain energy density per unit volume; W c is the critical strain energy density at which material damage occurs, which is a preset value; φ(F, ε ij , k) is the contribution function describing the contribution of the material microstructure to fatigue damage; g(ΔT, ω, c p ) is the temperature dissipation correction term.

[0025] Further, the calculation formula for the elastic strain energy density W s per unit volume is:

[0026]

[0027] The calculation formula for the contribution function φ(F, ε ij , κ) describing the contribution of the material microstructure to fatigue damage is:

[0028]

[0029] where F is the number of cycles; here, a cycle refers to a complete loading and unloading process experienced by the MEMS multi-layer composite structure under periodic loading; the periodic loading is the stress or strain that repeats in a mechanical or thermal field; F c is the maximum number of cycles; ΔF is the average number of cycles; the calculation formula for the temperature dissipation correction term g(ΔT, ω, c p ) is:

[0030]

[0031] where T c is the maximum allowable temperature; ω is the cycle frequency; ω c is the maximum cycle frequency.

[0032] Further, step 3: Set the following constraint conditions according to step 1 and step 2:

[0033]

[0034] where σ maxis the maximum allowable stress, which is a set value; D critical The maximum fatigue damage accumulation coefficient; h max is the maximum allowable thickness of each layer structure in the MEMS multi-layer composite structure; The objective function is expressed by the following formula:

[0035]

[0036] Among them, U is the optimal solution of the objective function; Ω represents the set of integral parameters composed of geometric parameters and material parameters.

[0037] The method for analyzing and optimizing the stress distribution of the MEMS structure in the micro-nano processing technology of the present invention has the following beneficial effects:

[0038] The present invention significantly improves the accuracy of stress analysis of the MEMS multi-layer composite structure. In traditional stress analysis, the model usually ignores the surface effect, interface effect and size effect at the micro-nano scale, and these factors have a key impact on the stress distribution in the multi-layer structure. In step 1 of the present invention, through the introduction of piezoelectric-elastic coupling and thermo-mechanical coupling effects, the elastic response, surface energy and temperature field of the material are comprehensively considered, and a more realistic stress field model is constructed. For example, the exponential decay term and interface shear stress term included in the formula can effectively capture the stress concentration phenomenon and interface behavior at the micro scale. This high-precision stress modeling method makes up for the limitations of traditional methods in the micro-nano environment and provides reliable data support for the precise design of MEMS devices. The present invention improves the accuracy of fatigue life prediction of the MEMS structure through the non-linear evaluation of fatigue damage. Traditional fatigue evaluation methods such as the Miner fatigue theory are usually based on the linear cumulative hypothesis and fail to fully consider the complex effects of material microstructure, temperature gradient and cyclic load frequency on fatigue behavior. In step 2 of the present invention, through the fatigue accumulation evaluation model, the modified stress field, the elastic strain energy density per unit volume, the temperature dissipation correction term and the material microstructure parameters are combined to propose a complete fatigue evaluation framework, which can capture the non-linear fatigue accumulation behavior under cyclic load and thermo-mechanical coupling. This comprehensive evaluation method greatly improves the accuracy of fatigue life prediction and provides guarantee for the long-term use of MEMS devices in complex working environments. Description of the Drawings

[0039] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only the embodiments of the present invention, and for those of ordinary skill in the art, other drawings can be obtained according to the provided drawings without creative work.

[0040] Figure 1Schematic flow diagram of the method for analyzing and optimizing the stress distribution of MEMS structures in the micro-nano processing technology provided by the embodiments of the present invention. Detailed implementation manners

[0041] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0042] Example 1: Refer to Figure 1 , the method for analyzing and optimizing the stress distribution of MEMS structures in the micro-nano processing technology, the method includes:

[0043] Step 1: Determine the geometric parameters and material parameters of each layer of the MEMS multi-layer composite structure, and construct the stress field of the MEMS multi-layer composite structure; based on the piezoelectric elastic coupling effect, establish a piezoelectric equation according to the stress field, and solve to obtain the strain tensor; according to the strain tensor, based on the thermo-mechanical coupling effect, establish a thermo-mechanical coupling equation, and solve the thermo-mechanical coupling equation to obtain the temperature distribution;

[0044] MEMS structures are often composed of multiple layers of materials stacked together. The physical properties of each layer of material, including elastic modulus, density, thermal expansion coefficient, piezoelectric coupling coefficient, etc., determine its complex coupling behavior under mechanical stress, thermal stress, and electric field. By extracting the geometric and physical characteristic parameters of each layer of material, a stress distribution model of the overall structure can be established. In this process, the piezoelectric elastic coupling effect is the first step in the analysis. Many materials in MEMS structures have piezoelectric properties, which means that mechanical strain and electric field affect each other. When stress acts on a piezoelectric material, an electric field is generated, and an applied electric field will cause mechanical deformation of the material. To accurately describe this phenomenon, it is necessary to relate the stress tensor of the material to physical quantities such as strain tensor and electric field strength through the piezoelectric equation. In the piezoelectric equation, the distribution of the stress field depends on the elastic properties of the material and the piezoelectric coupling coefficient, which reflect the sensitivity of the material to the coupling effect of mechanical and electric fields. By solving this equation, the strain tensor can be obtained, which describes the deformation distribution inside the material. The solution of the strain tensor is not only the basis for determining mechanical behavior but also an important input for subsequent thermo-mechanical coupling analysis.

[0045] Based on the strain tensor, the next step is to conduct a thermo-mechanical coupling effect analysis. In MEMS devices, due to phenomena such as the presence of electric current and mechanical energy release in the working environment, the thermal effect cannot be ignored. The thermo-mechanical coupling effect describes how thermal energy affects the stress distribution through heat conduction, thermal expansion, etc. Here, through the thermo-mechanical coupling equation, the strain tensor is introduced into the heat conduction model, and combined with parameters such as the thermal conductivity, density, and specific heat capacity of the material, a temperature field distribution model is established. This model takes into account the influence of the heat source distribution and boundary conditions on heat conduction, thus accurately predicting the temperature distribution inside the multi-layer structure. The distribution of the temperature field directly affects the thermal expansion behavior of the structure, and thermal expansion will further cause a redistribution of stress. Therefore, the temperature field analysis is the basis for the correction of the entire stress field. Through piezoelectric-elastic coupling and thermo-mechanical coupling analysis, the obtained stress field distribution not only reflects the mechanical stress, but also comprehensively considers the effects of the electric field and thermal effects. Compared with traditional methods, the innovation of the present invention lies in introducing an analysis framework of multi-field coupling, rather than being limited to the study of single-field effects. This multi-field coupling model is of great significance in actual MEMS design. MEMS devices are usually applied in complex environments, and a single mechanical or electric field model is difficult to reflect their true working state. By comprehensively modeling the piezoelectric effect and thermal effect, the present invention makes the prediction of the stress field distribution more accurate, thus providing a solid theoretical basis for subsequent optimization design. In addition, the present invention also emphasizes the flexibility and scalability of the stress field model. By adjusting the input parameters (such as geometric parameters, material properties, etc.), this method can adapt to different types of MEMS structures. This flexibility is crucial for practical applications because in the design and manufacturing process of MEMS devices, the selection of materials and changes in geometric structures will significantly affect their performance. By introducing the method of the present invention, the stress distribution under different design schemes can be quickly evaluated, providing a scientific basis for optimization design.

[0046] Step 2: According to the temperature distribution, correct the stress field to obtain the corrected stress field; according to the corrected stress field, use a preset fatigue cumulative assessment model to conduct fatigue assessment to obtain the fatigue damage cumulative coefficient;

[0047] The temperature field plays an important role in MEMS structures. Its distribution not only directly affects the expansion and contraction behavior of materials, but also has a profound impact on the overall structural performance by changing the local stress state. Based on the temperature distribution obtained in the first step, the initial stress field can be accurately corrected to obtain a modified stress field that better conforms to the actual working conditions, which lays a reliable foundation for subsequent fatigue damage assessment. The principle of correcting the stress field by temperature distribution is mainly based on the thermal expansion characteristics of materials. Since the MEMS multi-layer composite structure is composed of different materials, there may be significant differences in the thermal expansion coefficients of each layer of materials. Therefore, under the action of temperature changes, non-uniform expansion or contraction will occur between the material layers. This non-uniformity leads to stress concentration at the interfaces and may also trigger failure modes such as interface delamination and cracks. The temperature field obtained through the thermo-mechanical coupling effect can accurately describe the temperature changes at each position. Combining with the thermal expansion parameters of materials, the original stress field can be corrected so that the modified stress field can more realistically reflect the stress distribution state of the structure in the actual environment. Especially in high-precision MEMS devices, even the slightest stress changes caused by temperature can have a significant impact on the device performance. Therefore, this step plays an important role in stress distribution analysis and cannot be ignored.

[0048] After obtaining the corrected stress field, a fatigue cumulative assessment model is used to calculate the fatigue damage accumulation coefficient. Fatigue is one of the most common failure mechanisms in MEMS structures during long-term use, especially when subjected to cyclic loads or high-frequency vibrations. The fatigue performance of its materials directly determines the service life of the device. The fatigue cumulative assessment model is based on correlating the stress amplitude in the corrected stress field with the number of cyclic loads to evaluate the cumulative damage degree of the material at a specific stress level. This model is usually based on the mine fatigue theory or an improved linear cumulative damage model. By superimposing the contribution of each cyclic stress to the fatigue life, the overall fatigue damage accumulation coefficient is obtained. As an important indicator, the fatigue damage accumulation coefficient quantifies the fatigue state of the MEMS structure under the current working conditions and provides a clear reference basis for subsequent optimization design. It should be noted that in the present invention, the fatigue assessment model does not simply use the stress amplitude and the number of cycles as the only inputs, but performs dynamic calculations in combination with the corrected stress field. The corrected stress field not only includes the stress distribution under mechanical loads, but also comprehensively considers the influence of thermal effects on the stress, which makes the results of fatigue cumulative assessment more accurate. At the same time, due to the highly non-linear material properties in the MEMS multi-layer composite structure, traditional linear fatigue models may not be able to effectively describe complex fatigue behaviors. The present invention introduces an improved non-linear model to more comprehensively reflect the actual cumulative process of fatigue damage. The advantage of this method is that it not only realizes the dynamic correction of the stress field, but also provides a quantitative index through the fatigue cumulative assessment model to describe the fatigue state of the structure. This index can be directly used to evaluate the reliability of the MEMS structure and provide data support for structure optimization. For example, when it is detected that the fatigue damage accumulation coefficient of a certain material layer is too high, the thickness or material type of this layer of material can be adjusted, or even the stress concentration effect can be reduced by optimizing the structure design to extend the device life. This data-driven design optimization method is more scientific and targeted than traditional empirical design.

[0049] Step 3: Set the constraint conditions according to Step 1 and Step 2, construct an objective function based on the fatigue damage accumulation coefficient, and obtain the optimal solution of the objective function through iterative solution of the objective function; the optimal solution of the objective function includes the best geometric parameters and the best material parameters of the MEMS multi-layer composite structure; use the best geometric parameters and the best material parameters to optimize the design of the MEMS structure in the micro-nano processing technology.

[0050] In this step, it is first necessary to set the constraint conditions for the optimization process. These constraint conditions are derived from the analysis results of the previous two steps, including the corrected stress field and the fatigue damage accumulation coefficient. The corrected stress field provides the true stress distribution at each position in the MEMS structure, while the fatigue damage accumulation coefficient quantifies the fatigue state in different regions. By analyzing these results, the physical limitations that must be followed in the structural design can be determined. For example, the stress value in certain regions must not exceed the yield strength of the material, or the fatigue damage accumulation coefficient must be lower than a certain critical value to ensure the safety and reliability of the structure. These constraint conditions provide a boundary framework for the optimized design, ensuring the feasibility of the optimization results in practical applications.

[0051] Based on setting the constraint conditions, the next step is to construct the objective function. The objective function is the core of the optimized design, and its form directly determines the direction and result of the optimization. In the present invention, the objective function takes the geometric parameters and material parameters of the MEMS structure as design variables, comprehensively considering the uniformity of the stress distribution, the minimization of the fatigue damage accumulation coefficient, and the overall performance of the structure. By quantifying these factors into mathematical expressions, the objective function not only reflects the reliability of the structure but also fully considers functionality and economy. For example, in order to extend the service life of the structure, a greater weight can be assigned to the fatigue damage accumulation coefficient in the objective function. For a structure with high symmetry requirements, the uniformity of the stress distribution needs to be additionally considered. In addition, the objective function can be adjusted according to specific application scenarios to meet different design requirements. Solving the objective function is a highly complex process because the design variables of the MEMS multi-layer composite structure often have high dimensionality and non-linearity. In the present invention, an iterative solution method is used to find the optimal solution of the objective function. The key to iterative solution lies in selecting a suitable optimization algorithm, such as the gradient descent method, genetic algorithm, or particle swarm optimization algorithm, etc. These algorithms can efficiently search in high-dimensional space and avoid getting trapped in local optimal solutions. Taking the genetic algorithm as an example, it gradually optimizes the combination of design variables by simulating the genetic variation and selection mechanisms in nature, so as to find the parameters that make the objective function reach the optimum. The selection and parameter setting of the optimization algorithm have an important impact on the quality of the results and the computational efficiency, so they need to be reasonably designed in combination with the actual problem.

[0052] By iteratively solving the objective function, the optimal combination of geometric and material parameters of the MEMS structure can be obtained. These optimization results not only satisfy the constraint conditions but also significantly improve the performance of the structure. For example, the optimized geometric parameters can reduce the stress concentration effect, thereby reducing the cumulative rate of local fatigue damage; the optimized material parameters can improve the overall strength and stability of the structure, thus adapting to more demanding working environments. Compared with traditional design methods, the present invention greatly improves the scientificity and accuracy of the design through the quantitative description of the objective function and the iterative optimization process. Step 3 of the present invention is significantly innovative technically. Traditional MEMS structure designs usually rely on experience and experiments, lack systematic optimization methods, and it is difficult to fully consider multi-field coupling effects and fatigue behavior. The present invention combines stress distribution, fatigue assessment, and optimization design to form a complete closed loop, making the design process more comprehensive and efficient. In addition, the objective function and optimization algorithm proposed by the present invention have high flexibility and can adapt to different types of MEMS structures and application scenarios. This feature makes this method not only applicable to current MEMS designs but also has strong foresight and can provide theoretical and method support for future more complex micro-nano structure designs.

[0053] Example 2: The geometric parameters of each layer structure include: layer thickness, material characteristic length, and stress attenuation length; the material parameters of each layer structure include: Young's modulus, Poisson's ratio, surface energy, and interfacial shear stress.

[0054] Specifically, the geometric parameters of each layer structure include layer thickness, material characteristic length, and stress attenuation length. These parameters are the core elements for describing the geometric characteristics of MEMS structures. The layer thickness refers to the vertical dimension of each material layer and has a direct impact on the stress distribution. In a multi-layer composite structure, the layer thickness determines the deformation coordination of different material layers and the non-uniformity of thermal expansion. For example, a thicker layer has higher rigidity when subjected to external forces, but it may also cause more severe stress concentration. The material characteristic length is usually used to describe the characteristic size of the structure at the microscale, such as the grain size or the typical length of the substructure, and it directly affects the elastic and plastic behavior of the material. At the micro-nano scale, the characteristic length has a particularly significant impact on the structural performance because many mechanical and thermodynamic properties exhibit obvious scale effects as the size decreases. The stress attenuation length is another important parameter, which reflects the propagation and attenuation characteristics of stress in the material. The shorter the stress attenuation length, the more limited the range of action of the external force, which is crucial for predicting local stress distribution and interface failure. The material parameters of each layer structure include Young's modulus, Poisson's ratio, surface energy, and interface shear stress. These parameters together determine the mechanical properties and interface behavior of the material. Young's modulus is an important indicator for measuring the stiffness of the material and is used to describe the elastic deformation ability of the material under external forces. In MEMS structures, the difference in Young's modulus between different material layers significantly affects the stress distribution, especially stress concentration may occur at the interface. Poisson's ratio describes the ratio of the transverse deformation to the axial deformation of the material during tension or compression, and it, together with Young's modulus, determines the elastic behavior of the material. For example, a material with a larger Poisson's ratio will exhibit significant transverse expansion when compressed, which will further affect the coordinated deformation of the multi-layer structure. The surface energy is a key parameter at the micro-nano scale and reflects the binding energy of the atoms on the material surface. In MEMS structures, changes in surface energy not only affect the mechanical properties of the material but also play an important role in the adhesion and slip behavior at the interface. The interface shear stress is used to describe the relative sliding ability of two layers of materials at the interface and is the core parameter for interface failure analysis. The larger the interface shear stress, the stronger the bonding between the material layers, but it may also lead to greater residual stress. Combining these geometric parameters and material parameters, the present invention can more accurately describe the mechanical behavior of the material layer and the coupling characteristics of the multi-layer structure in the stress field analysis of MEMS structures. In the modeling stage, these parameters are directly used to construct the stress field and thermo-mechanical coupling equations, providing a high-quality initial data source for subsequent correction and optimization. For example, the layer thickness and Young's modulus directly affect the solution results of the stress field, while the stress attenuation length and interface shear stress determine the details of the interface stress distribution. In addition, in the optimization design, these parameters are incorporated into the objective function as design variables, and by adjusting their values, the performance of the structure can be effectively improved. For example, by optimizing the combination of surface energy and Poisson's ratio, the stress concentration effect at the interface can be reduced, thereby improving the reliability of the overall structure.

[0055] Example 3: In step 1, the stress field of the MEMS multi-layer composite structure is expressed by the following formula:

[0056]

[0057] where Q n is the Young's modulus of the nth layer structure; v n is the Poisson's ratio of the nth layer structure; γ n is the surface energy of the nth layer structure; h n is the layer thickness of the nth layer structure; l0 is the material characteristic length, which is a preset value; l c is the stress attenuation length, which is a preset value; τ int (x, y) is the interfacial shear stress at the position (x, y); x is the X-axis coordinate of the two-dimensional plane; y is the Y-axis coordinate of the two-dimensional plane; z is the Z-axis coordinate of the three-dimensional space; z int is the interface position; δ(·) is the Dirac function; σ total is the stress field; N is the number of layers of the MEMS multi-layer composite structure.

[0058] Specifically, the total expression of the stress field is obtained by superimposing the stress contributions of all layers. This way of layer-by-layer summation essentially reflects the different influences of each layer of material on the overall stress field. In the formula, the elastic stress term is reflected by the combination of Young's modulus and Poisson's ratio, specifically shown as This form emphasizes the elastic response characteristics of the material, where the Young's modulus Q n determines the stiffness of the material, while the Poisson's ratio v n describes the lateral deformation ability of the material. The combination of the two ensures that the elastic stress distribution can truly reflect the mechanical properties of the material. This part of the description is crucial for multi-layer composite structures because differences in Young's modulus and Poisson's ratio of different materials may lead to significant stress concentration or uneven distribution phenomena, and the formula precisely captures these complex effects through this term. In addition to the influence of elastic stress, the formula also particularly focuses on the contribution of surface effects to stress. Through the surface stress term the surface energy γ n and the layer thickness h n are introduced into the model. In micro-nano structures, the role of surface energy accounts for a significant proportion in the total stress because the binding force between surface atoms has a particularly prominent impact on the mechanical behavior of materials at small scales. The layer thickness h n as an inverse proportional factor reflects the law that the smaller the thickness, the more significant the surface effect. In addition, the introduction of the material characteristic length l0 further enhances the formula's ability to depict micro-scale size effects. This description of surface stress is particularly important in MEMS structure design because it can reveal the influence of different layer thicknesses and material selections on the overall stress distribution, providing key data support for optimized design.

[0059] The spatial distribution characteristics of the stress field are described by an exponential decay term in the form of This term characterizes the behavior of the stress gradually decaying as the distance z increases. The decay length l c is a parameter related to the material properties, indicating the propagation range of the stress in the material. By means of exponential decay, the stress distribution inside the material can be accurately simulated, especially the stress concentration phenomenon near the interface. This local stress description is of great significance in the MEMS multi-layer composite structure because the stress concentration at the interface is often one of the main reasons for structural failure. The last part of the formula describes the influence of the interfacial shear stress through τ int (x, y)δ(z - z int ). Among them, τ int (x, y) represents the shear stress distribution at the interface, which reflects the interaction between material layers in the multi-layer structure. This term particularly considers the particularity of the interface position (z = z int ) and uses the Dirac function δ(z - z int ) to centralize the shear stress at the interface. The interfacial stress plays a key role in the multi-layer structure because the bonding strength and relative sliding ability between different materials directly affect the stability and reliability of the overall structure. Through this term, the local deformation or failure behavior caused by shear stress at the interface can be accurately simulated, thus providing a theoretical basis for interface design and optimization. Overall, the present invention ingeniously integrates the elastic properties, surface effects, and interfacial behavior of materials into a unified mathematical model in the stress field formula. Compared with traditional stress analysis methods, this formula has significant innovation because it can not only accurately describe the stress distribution of the multi-layer composite structure but also adapt to different types of MEMS structures and application scenarios by adjusting the model parameters. For example, by adjusting the layer thickness h n and the material characteristic length l0, the influence of the size effect on the stress distribution can be studied; by changing the interfacial shear stress τ int (x, y), the bonding performance of the interface can be optimized. This parametric analysis method significantly improves the flexibility and applicability of stress field modeling.

[0060] Example 4: The piezoelectric equation in Step 1 is expressed by the following formula:

[0061]

[0062] Among them, i, j, k, l, and m are all subscript indices representing directions, and their values are all 1 or 2; when both i and j have a value of 1, it represents the normal strain in the X-axis direction; when i has a value of 1 and j has a value of 2, it represents the shear strain in the two-dimensional plane formed by the X-axis and the Y-axis; when both k and l have a value of 1, it represents the normal stress in the X-axis direction; when k has a value of 1 and l has a value of 2, it represents the shear stress in the two-dimensional plane formed by the X-axis and the Y-axis; when m has a value of 1, the electric displacement is in the X-axis direction; when m has a value of 2, the electric displacement is in the Y-axis direction; D m is the electric displacement vector; S ijkl is the compliance coefficient; ε ij is the strain tensor, which is a symmetric quantity, ε ij = ε ji ; d mij are all piezoelectric strain coefficients; represents the divergence of D m ; represents the second-order gradient of σ total ; α cp is the micro-nano scale correction coefficient, which is a preset value; ∈ mk represents the dielectric coefficient; E m is the electric field strength vector.

[0063] Example 5: The thermo-mechanical coupling equation in Step 1 is calculated using the following formula:

[0064]

[0065] Among them, ρ is the density of the MEMS multi-layer composite structure; c p is the specific heat capacity of the MEMS multi-layer composite structure; T is the temperature distribution; t is the time; is the divergence calculation; represents the gradient of the temperature distribution; κ is the thermal conductivity of the MEMS multi-layer composite structure; α T is the preset thermo-mechanical coupling coefficient; η is the preset energy dissipation coefficient; M is the mass of the MEMS multi-layer composite structure.

[0066] Specifically, the left side of the piezoelectric equation represents the response of the system, including the strain tensor ε ij and the electric displacement vector D m , which are the direct manifestation forms of the multi-layer structure under the action of external mechanical loads and electric fields. The strain tensor ε ij describes the deformation of the material under the action of external forces, where i and j are direction subscripts, and through combination, it can represent normal strain (deformation along the axis) and shear strain (deformation in the two-dimensional plane). The electric displacement vector D mdescribes the electrode polarization phenomenon of the material under the action of an electric field, representing the polarization effect generated by the change in charge distribution inside the material. The first term on the right side is the elastic-piezoelectric coupling part, through the compliance coefficient matrix S ijkl , the piezoelectric strain coefficient d mij and the dielectric constant matrix ∈ mk relate the relationships among stress, strain, and electric field strength. The compliance coefficient S ijkl is the basic mechanical parameter of the material, determining the linear relationship between stress and strain. The piezoelectric strain coefficient d mij describes the coupling degree between mechanical behavior and electrical behavior, which reflects the piezoelectric response intensity of the material when subjected to force or an electric field. The dielectric constant matrix ∈ mk then describes the electrical properties of the material, including the distribution of the electric field inside the material and the polarization effect. This part mainly reflects the description ability of the macroscopic piezoelectric effect and is applicable to MEMS structures at the conventional scale.

[0067] The second term on the right side is the micro-nano scale correction part, through the correction coefficient α cp and the high-order gradient term and consider the micro-nano scale effect. These terms are the innovative points of the present invention, aiming to capture the nonlinear and high-order effects that cannot be explained by the traditional piezoelectric theory in micro-nano structures. At the micro-nano scale, the mechanical and electrical behaviors of the material will be significantly affected by surface effects, interface effects, and size effects. The correction coefficient α cp is an empirical parameter used to quantify the influence of micro-nano scale effects on the overall response of the system. The second-order gradient of the stress field and the divergence of the electric displacement The local rates of change of the stress field and the electric displacement field are described separately. These higher-order terms can capture local non-uniformities and coupling behaviors, such as stress concentration at the interface, changes in surface charge distribution, etc. The coupling characteristics in the formula reflect the complex interaction between the mechanical behavior and the electrical behavior in the MEMS multi-layer composite structure. The strain tensor is connected to the total stress field through the compliance coefficient, and the total stress field affects the electric displacement field and the electric field strength through the piezoelectric strain coefficient and the dielectric constant. This multi-field interaction relationship makes the formula highly comprehensive and universal, applicable not only to traditional piezoelectric materials but also capable of effectively describing the behaviors of new multifunctional materials and composite materials at the micro-nano scale. From the perspective of the subject matter of the present invention, this piezoelectric equation provides a theoretical support for the multi-field coupling analysis in MEMS structures and has significant innovation compared with the prior art. Traditional piezoelectric equations usually only consider the linear relationship between elastic behavior and electrical behavior, ignoring the complex effects at the micro-nano scale. However, the present invention significantly improves the accuracy and applicability of the model by introducing micro-nano scale correction coefficients and high-order gradient terms. In addition, this equation, in a parameterized form, can adapt to various materials and geometric structures, providing a scientific basis for the design and optimization of complex MEMS structures. For example, by adjusting the compliance coefficient, piezoelectric strain coefficient, and correction coefficient, the influence of different design parameters on the structural performance can be predicted, providing data support for the optimization design.

[0068] Example 6: In step 2, according to the temperature distribution, the stress field is corrected through the following formula to obtain the corrected stress field σ modified :

[0069]

[0070] where T0 is the initial temperature; λ th is the preset thermo-sensitive coefficient.

[0071] Specifically, the first core term in the formula is the total stress field σ total , which is obtained from the basic stress distribution model constructed in the previous step 1. This term serves as the initial value and provides a starting point for subsequent corrections. The total stress field essentially describes the stress distribution of the structure at room temperature and under no thermal load conditions, and its accuracy directly affects the reliability of the corrected stress field. Before the temperature influence is added, σ total is mainly determined by mechanical loads and interface effects. Therefore, the significance of the correction process is to introduce temperature into the model to be closer to the actual working environment. The temperature influence is manifested through the second term of the formula. The physical meaning of this term is the thermal stress correction amount caused by temperature change, where each layer of material in the multi-layer composite structure contributes to the overall stress field through summation one by one. Here, reflects the elastic modulus Q of the materialn and Poisson's ratio v n The coupled effect on thermal stress describes the material's response ability under thermal expansion or contraction conditions. The temperature difference (T - T0) is the driving force for this term, representing the change in the current temperature compared to the initial temperature, reflecting the direct impact of the external thermal environment on the material properties. At the same time, this term also introduces the surface effect of the material where the surface energy γ n and the layer thickness h n together determine the intensity of the thermal stress correction. The surface effect is particularly important in micro - nano structures because as the structure size decreases, the binding force between surface atoms has a significantly enhanced impact on the mechanical behavior. In addition, the exponential term further reflects the attenuation effect of the temperature gradient on the thermal stress. This term indicates that when the temperature gradient is large, the contribution of local thermal stress will rapidly weaken, reflecting the complex influence of heat diffusion and local temperature difference on stress. This non - linear description enhances the applicability of the formula in dealing with complex thermal environments. The third term is the thermodynamics diffusion term, used to capture the deeper influence of the temperature distribution on the internal stress of the material. The thermo - sensitivity coefficient λ th is a parameter related to the material properties, used to quantify the expansion effect of the temperature field on the stress distribution. By introducing the second - order gradient of the temperature field the formula can more comprehensively describe the conduction and distribution behavior of temperature changes inside the material. Especially in MEMS structures, due to the complex changes in thermal conductivity and thermal expansion coefficient at the micro - scale, this term is crucial for improving the accuracy of the model. Generally speaking, through the combination of three parts, this formula completes the complete modeling from the total stress field to the corrected stress field. Compared with traditional stress analysis methods, the formula of the present invention has significant innovation. First, by comprehensively considering elasticity, surface effect and temperature gradient, the formula can more comprehensively describe the complex thermodynamics behavior in multi - layer composite structures. Second, the introduction of the exponential term and high - order gradient makes the formula more precise in dealing with micro - scale effects and local non - uniformity. This feature is of great significance for the design and optimization of MEMS devices because traditional models often have difficulty effectively capturing the non - linear effects of multi - field coupling in micro - nano structures

[0072] Example 7: In step 2, according to the corrected stress field, use the preset fatigue cumulative evaluation model to conduct fatigue evaluation through the following formula to obtain the fatigue damage cumulative coefficient D:

[0073]

[0074] where b is the preset material fatigue index, and the smaller the b value, the easier the MEMS multi - layer composite structure is to be fatigued by stress; σ′ fis the fatigue strength constant of the preset MEMS multi-layer composite structure, with a value range of 20 to 30; W s is the elastic strain energy density per unit volume; W c is the critical strain energy density at which material damage occurs, which is a preset value; φ(F, ε ij , κ) is the contribution function describing the contribution of the material microstructure to fatigue damage; g(ΔT, ω, c p ) is the temperature dissipation correction term.

[0075] Specifically, the integral form of the formula indicates that fatigue damage accumulation is a process that changes dynamically with time. The upper limit of the integral is time t, representing the overall calculation of the cumulative effect of fatigue behavior within the working cycle. The expression under the integral is divided into several terms, and each term describes the contribution of different physical mechanisms to fatigue damage. The first part of the formula is the fatigue stress contribution term based on the modified stress field. The modified stress field σ modified has comprehensively considered various influences such as temperature, surface effect, and geometric characteristics through the previous steps, and can truly reflect the stress state of the material under complex loads. Comparing σ modified with the fatigue strength constant σ′ of the material f and raising the ratio to the exponent b is used to quantify the relationship between the stress level and fatigue accumulation. The fatigue exponent b is a parameter related to the material properties. The smaller its value, the more sensitive the material is to stress, so significant fatigue damage will occur at a lower stress level. This part reveals the fatigue response of the material under different stress conditions and captures the non-linear characteristics through the exponential form. The second part is the elastic energy damage factor based on the elastic strain energy density W per unit volume s and the critical strain energy density W. The elastic strain energy density W s represents the mechanical energy stored per unit volume, while the critical strain energy density W c is the energy limit at which material damage occurs. Through the exponential function form, when W s approaches or exceeds W c , this term will increase rapidly, indicating that the fatigue damage accumulation enters the rapid growth stage at this time. This part effectively captures the mutational characteristics of fatigue damage, that is, the rapid failure behavior of the material when approaching the damage limit. The third part φ(F, ε ij , k) is the contribution function describing the influence of the material microstructure. This function comprehensively considers the external force F, the strain tensor ε ij and the microstructure parameter k of the material. The external force F is the direct load factor, and the strain tensor ε ijdescribes the local deformation of the material under external forces, while the microstructure parameter κ reflects the microscopic characteristics inside the material, such as grain size, porosity, etc. The importance of this term lies in that incorporating the microscopic structure characteristics of the material into the fatigue damage modeling can reflect the essential influence of the internal structures of different materials on fatigue behavior, thus making the model more physically meaningful. The last part g(ΔT, ω, c p ) is the temperature dissipation correction term, which is used to describe the regulating effect of thermal effects on fatigue damage accumulation. The temperature difference ΔT, frequency ω, and specific heat capacity c p are the key parameters of the temperature field, which respectively represent the temperature fluctuation range, the application frequency of the load, and the material's response ability to heat. Temperature fluctuations have a direct impact on fatigue damage in the form of thermal stress, while high-frequency loads may cause local heat accumulation, thus accelerating fatigue damage accumulation. The specific heat capacity c p determines the thermal buffering ability of the material under thermal loads. Therefore, g(ΔT, ω, c p ) synthesizes the correction effects of these factors on fatigue accumulation. Combining the overall framework of the formula, the fatigue assessment model proposed in the present invention introduces more physical mechanisms and material characteristics on the basis of the traditional model, making the calculation of fatigue damage accumulation more in line with the actual working conditions. Compared with the prior art, the present invention not only considers the non-linear relationship between stress level and fatigue behavior, but also further improves the accuracy and applicability of the model through elastic strain energy and microstructure parameters. At the same time, the addition of the temperature dissipation correction term enables the model to effectively describe the influence of complex thermal environments on fatigue accumulation, which is particularly important in the design of MEMS devices in high-frequency vibration and large temperature difference environments.

[0076] Example 8: The elastic strain energy density W s per unit volume is calculated as follows:

[0077]

[0078] The contribution function φ(F, ε ij , κ) describing the contribution of the material microstructure to fatigue damage is calculated as follows:

[0079]

[0080] where F is the number of cycles; here, a cycle refers to a complete loading and unloading process experienced by the MEMS multi-layer composite structure under cyclic loads; cyclic loads are stresses or strains that act repeatedly in a mechanical or thermal field; F c is the maximum number of cycles; ΔF is the average number of cycles; the calculation formula for the temperature dissipation correction term g(ΔT, ω, c p ) is as follows:

[0081]

[0082] Among them, T c is the maximum allowable temperature; ω is the cycle frequency; ω c is the maximum cycle frequency.

[0083] Specifically, the elastic strain energy density W s The calculation formula It quantitatively describes the mechanical energy stored in a unit volume due to the modified stress field and strain tensor. The modified stress field σ in the formula modified It is the stress distribution after the combined influence of temperature, surface effect and geometric characteristics, which can truly reflect the stress state of the material under complex loads. ij describes the degree of deformation of a material under the action of an external force. This formula directly relates stress and strain, capturing the energy storage properties of the material within the elastic range. Elastic strain energy density is an important factor affecting fatigue damage, because as strain energy accumulates, the microstructure inside the material gradually deteriorates, eventually leading to fatigue failure. Fatigue damage contribution function φ(F, ε ij ,κ) further reveals the influence of material microstructure and load cycle on fatigue behavior. In the formula, It reflects the inhibitory effect of the microstructure parameter κ on damage. Microstructure parameters are usually related to the grain size, porosity or interface characteristics of the material. The larger the value, the lower the fatigue resistance of the material. Strain tensor The exponential b describes the nonlinear effect of strain amplitude on fatigue damage, and the exponential form enhances the model's ability to capture the sensitivity of strain to fatigue accumulation. In addition, Through the maximum number of cycles F c The average number of cycles ΔF simulates the accumulation of fatigue damage during the load cycle. This part reflects the time dependence of fatigue damage accumulation, indicating that the fatigue growth of the material is slow in the initial cycle, but as the number of cycles approaches the limit, fatigue damage will accumulate rapidly. The temperature dissipation correction term g(ΔT, ω, c p ) describes the correction effect of thermal environment on fatigue damage. In the formula, Indicates the negative impact of temperature fluctuation ΔT on fatigue accumulation. Maximum allowable temperature T c It is the thermal limit of the material under working conditions. When ΔT approaches or exceeds T c Temperature fluctuations can significantly accelerate the accumulation of fatigue damage. Combined with the specific heat capacity c p , cycle frequency ω and maximum cycle frequency ω cThe influence. High-frequency loads can trigger local heat accumulation, thereby exacerbating material damage. This part effectively captures the regulatory effect of thermodynamics on fatigue behavior in a non-linear form. Overall, these formulas systematically describe the complex mechanism of fatigue damage through multi-level modeling. Compared with traditional fatigue analysis models, the present invention has significant innovations in several key aspects. First, through the elastic strain energy density W s , the model can accurately quantify the mechanical energy accumulation inside the material, providing a reliable physical basis for fatigue prediction. Second, the fatigue damage contribution function φ(F, ε ij , k) not only considers the non-linear influence of the strain level, but also reflects the dynamic characteristics of fatigue failure through the cumulative process of the number of cycles. Finally, the introduction of the temperature dissipation correction term g(ΔT, ω, c p ) significantly improves the applicability of the model in complex thermal environments. Especially in the design of MEMS structures under high-frequency loads and large temperature differences, this part can effectively predict the influence of thermo-mechanical coupling on fatigue behavior.

[0084] Example 9: Step 3: Set the following constraint conditions according to Step 1 and Step 2:

[0085]

[0086] where σ max is the maximum allowable stress, which is a set value; D critical is the maximum fatigue damage accumulation coefficient; h max is the maximum allowable thickness of each layer in the MEMS multi-layer composite structure; the objective function is expressed by the following formula:

[0087]

[0088] where U is the optimal solution of the objective function; Ω represents the set of integral parameters composed of geometric parameters and material parameters.

[0089] Specifically, first, the starting point of the optimization process lies in the setting of constraint conditions, which mathematically define the boundaries of the design space and physically limit the unreasonable solutions during the optimization process. Among them, the first constraint condition is based on the energy balance expression of the modified stress field and microstructure. By introducing a correction coefficient and a quadratic form description of the strain energy, it ensures the mechanical balance of the multi-layer composite structure under complex loads. The core of the mechanical balance is that the modified stress field reflects the true state of the structure in the actual usage environment, which comprehensively incorporates the complex effects of temperature gradients, surface effects, and multi-layer geometric characteristics. Therefore, the mechanical balance is not only a theoretical requirement but also the key to ensuring the engineering feasibility of the optimization design results. Further, the strain energy description in the formula explicitly incorporates the non-linear behavior of the microstructure into the balance equation, enabling the formula to capture micro-scale effects that are difficult to explain by traditional models, such as local deformation and stress concentration at the interfaces. The second constraint condition limits the maximum value of the total stress field, which directly reflects the strength limit requirement of the material. The total stress field is determined by the coupling of material properties, geometric parameters, and external fields such as temperature in the multi-layer structure. The physical meaning of σ total ≤σ max is to prevent local stress from exceeding the yield strength of the material, thereby causing plastic deformation or structural failure. By adding this condition, the optimization process ensures that the mechanical properties of each layer of material and the overall structure are within the safe operating range, which is crucial for the design of high-precision and high-reliability MEMS devices. The third constraint condition limits the fatigue damage accumulation coefficient, specifying that the calculated cumulative value D in the fatigue assessment model must be less than the preset fatigue tolerance D critical . This condition is set based on the fatigue characteristics of the MEMS multi-layer structure under long-term cyclic loads. The fatigue accumulation process is a time-dependent dynamic behavior. By restricting the accumulation rate of D, the service life of the structure can be effectively extended. In addition, this condition also reflects the high attention of the optimization design to the structural reliability, ensuring that the final designed MEMS device can operate stably in the long term, especially under high-frequency vibration or extreme environments. The fourth constraint condition limits the layer thickness. By restricting the thickness h n of each layer not to exceed the maximum value h max , the optimization design process achieves a balance in terms of structural strength, mass, heat dissipation performance, etc. The thickness distribution in the multi-layer structure directly affects the uniformity of the stress distribution and the strength of the interface effect. Therefore, the limitation of the layer thickness is not only a basic requirement for geometric design but also affects key characteristics such as fatigue performance and heat conduction efficiency.

[0090] Based on the set constraints, the objective function quantifies and weighs multiple indicators of the optimal design in an integral form. The objective function includes core parameters in three aspects: fatigue damage, stress field, and temperature distribution, and assigns corresponding importance to different indicators in a weighted form. For example, the fatigue damage term 0.2*D reflects the degree of influence of fatigue accumulation with a relatively low weight, indicating that the goal of optimization is to control the growth rate of fatigue damage rather than completely eliminate the fatigue phenomenon. The stress field term emphasizes the attention of the optimal design to the stress distribution with a relatively high weight because local stress concentration is often one of the main causes of failure. The temperature term incorporates the influence of the thermal field into the optimization model through the normalization of the temperature distribution. Especially under high-temperature or large temperature difference conditions, this part can significantly improve the thermal stability of the MEMS structure. The integration region Ω of the objective function represents the design space composed of geometric parameters and material parameters. By optimizing and searching in this high-dimensional space, the solution U that minimizes the value of the objective function is found. The mathematical essence of this process is the solution of a multi-objective optimization problem, and its core difficulty lies in the non-linearity and high-dimensionality of the objective function. Therefore, advanced optimization algorithms such as genetic algorithms, particle swarm optimization, or gradient descent methods are usually required to gradually approach the optimal solution through iteration. Overall, the constraints and objective function of Example 9 together constitute a complete optimization framework, systematically solving the complex problem of multi-physical field coupling in the design of MEMS multi-layer composite structures. Compared with traditional design methods, the present invention has significant innovations in the following aspects. First, the setting of constraints is based on the comprehensive analysis of physical fields and geometric characteristics, which can effectively avoid the emergence of invalid solutions and engineering-infeasible solutions. Second, the weighted form of the objective function reflects the competition and trade-off between design indicators, and the best design solution that takes into account both reliability and performance can be found through the optimization algorithm. Finally, the optimization framework of the present invention has high flexibility and scalability, and can adapt to the design requirements of MEMS devices with different materials, geometric structures, and working conditions.

[0091] The present invention has been introduced in detail above. Specific examples are used in this article to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention. It should be noted that for those of ordinary skill in the art in this technical field, without departing from the principle of the present invention, several improvements and modifications can be made to the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. The stress distribution analysis and optimization method of MEMS structure in micro-nano processing technology is characterized by: The method comprises: Step 1: Determine the geometric parameters and material parameters of each layer of the MEMS multilayer composite structure, and construct the stress field of the MEMS multilayer composite structure; based on the piezoelectric elastic coupling effect, establish the piezoelectric equation according to the stress field, and solve it to obtain the strain tensor; based on the strain tensor and the thermomechanical coupling effect, establish the thermomechanical coupling equation, and solve the thermomechanical coupling equation to obtain the temperature distribution; Step 2: According to the temperature distribution, the stress field is corrected to obtain a corrected stress field; according to the corrected stress field, a fatigue assessment is performed using a preset fatigue accumulation assessment model to obtain a fatigue damage accumulation coefficient; Step 3: Set constraints according to steps 1 and 2, construct an objective function according to the fatigue damage accumulation coefficient, and obtain the optimal solution of the objective function by iteratively solving the objective function; the optimal solution of the objective function includes the optimal geometric parameters and optimal material parameters of the MEMS multilayer composite structure; use the optimal geometric parameters and optimal material parameters to optimize the design of the MEMS structure in the micro-nano processing technology.

2. The method for analyzing and optimizing stress distribution of MEMS structures in micro-nano processing technology according to claim 1, characterized in that: The geometric parameters of each layer structure include: layer thickness, material characteristic length and stress attenuation length; the material parameters of each layer structure include: Young's modulus, Poisson's ratio, surface energy and interface shear stress.

3. The method for analyzing and optimizing stress distribution of MEMS structures in micro-nano processing technology according to claim 2, characterized in that: In step 1, the stress field of the MEMS multilayer composite structure is expressed using the following formula: Among them, Q n is the Young's modulus of the n-th layer structure; v n is the Poisson's ratio of the n-th layer structure; n is the surface energy of the nth layer structure; h n is the layer thickness of the nth layer structure; l0 is the material characteristic length, which is the preset value; l c is the stress decay length, which is the preset value; τ int (x, y) is the interfacial shear stress at position (x, y); x is the X-axis coordinate of the two-dimensional plane; y is the Y-axis coordinate of the two-dimensional plane; z is the Z-axis coordinate of the three-dimensional space; int is the interface position; δ(·) is the Dirac function; σ total is the stress field; N is the number of layers of the MEMS multilayer composite structure.

4. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 3, characterized in that: The piezoelectric equation in step 1 is expressed as follows: Among them, i, j, k, l and m are all subscript indexes indicating directions, and their values ​​are all 1 or 2; when i and j are both 1, it indicates the normal strain along the X-axis direction; when i is 1 and j is 2, it indicates the shear strain along the two-dimensional plane formed by the X-axis and the Y-axis; when k and l are both 1, it indicates the normal stress along the X-axis direction; when k is 1 and l is 2, it indicates the shear stress along the two-dimensional plane formed by the X-axis and the Y-axis; when m is 1, the electric displacement is along the X-axis direction; when m is 2, the electric displacement is along the Y-axis direction; D m is the electric displacement vector; S ijkl is the flexibility coefficient; ε ij is the strain tensor, a symmetric quantity, ε ij =ε ji ;d mij All are piezoelectric strain coefficients; Indicates D m The divergence of Represents σ total The second-order gradient of cp is the micro-nano scale correction coefficient, which is the preset value; ∈ mk Represents the dielectric constant; E m is the electric field strength vector.

5. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 4, characterized in that: The thermomechanical coupling equation in step 1 is calculated using the following formula: Where ρ is the density of the MEMS multilayer composite structure; c p is the specific heat capacity of the MEMS multilayer composite structure; T is the temperature distribution; t is the time; is the divergence calculation; represents the gradient of temperature distribution; k is the thermal conductivity of the MEMS multilayer composite structure; α T is the preset thermal coupling coefficient; η is the preset energy dissipation coefficient; M is the mass of the MEMS multilayer composite structure.

6. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 5, characterized in that: In step 2, the stress field is corrected according to the temperature distribution using the following formula to obtain the corrected stress field σ modified : Where T0 is the initial temperature; λ th is the preset thermal sensitivity coefficient.

7. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 6, characterized in that: In step 2, fatigue damage accumulation coefficient D is obtained by using the following formula based on the corrected stress field and the preset fatigue accumulation evaluation model to perform fatigue evaluation: Where b is the preset material fatigue index. The smaller the b value is, the more susceptible the MEMS multilayer composite structure is to fatigue due to stress. f is the fatigue strength constant of the preset MEMS multilayer composite structure, ranging from 20 to 30; W s is the elastic strain energy density per unit volume; W c is the critical strain energy density for material damage, which is a preset value; φ(F, ε ij , k) is the contribution function of the material microstructure to fatigue damage; g(ΔT, ω, c p ) is the temperature dissipation correction term.

8. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 7, characterized in that: Elastic strain energy density per unit volume W s The calculation formula is: Function φ(F, ε) describing the contribution of material microstructure to fatigue damage ij , k) is calculated as: Where F is the number of cycles; where a cycle refers to a complete loading and unloading process that a MEMS multilayer composite structure undergoes under a periodic load; a periodic load is a stress or strain that acts repeatedly in a mechanical or thermal field; F c is the maximum number of cycles; ΔF is the average number of cycles; the temperature dissipation correction term g(ΔT, ω, c p ) is calculated as: Among them, T c is the maximum allowable temperature; ω is the cycle frequency; ω c is the maximum cycle frequency.

9. The method for analyzing and optimizing stress distribution of MEMS structure in micro-nano processing technology according to claim 8, characterized in that: Step 3: Set the following constraints based on steps 1 and 2: Among them, σ max is the maximum allowable stress, is the set value; D critical Maximum fatigue damage accumulation factor; h max is the maximum allowable thickness of each layer in the MEMS multilayer composite structure; the objective function is expressed using the following formula: Among them, U is the optimal solution of the objective function; Ω represents the integral parameter set composed of geometric parameters and material parameters.

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